OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 0 = Sum_{n>=1} n * ((1+x)^n - A(x))^n.
(2) A(x) = P(x)/Q(x) where
P(x) = Sum_{n>=0} n * (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+2),
Q(x) = Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n*A(x))^(n+2).
(3) A'(x) = P(x)/Q(x) where
P(x) = Sum_{n>=0} (n+1)^3 * ((1+x)^(n+1) - A(x))^n * (1+x)^n,
Q(x) = Sum_{n>=0} (n+1)^2 * ((1+x)^(n+1) - A(x))^n.
a(n) ~ c * d^n * sqrt(n) * n!, where d = A317855 = 3.16108865386... and c = 0.102568345138... - Vaclav Kotesovec, Jun 05 2019
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 20*x^3 + 282*x^4 + 5134*x^5 + 112053*x^6 + 2823119*x^7 + 80202565*x^8 + 2529045393*x^9 + 87523776013*x^10 + ...
such that
0 = ((1+x) - A(x)) + 2*((1+x)^2 - A(x))^2 + 3*((1+x)^3 - A(x))^3 + 4*((1+x)^4 - A(x))^4 + 5*((1+x)^5 - A(x))^5 + 6*((1+x)^6 - A(x))^6 + ...
The terms a(n) modulo 2 begin:
1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,
0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,
0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,1,1,
0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,
1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,1,
0,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,1,1,1,1,0,0,
1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,
1,1,0,0,0,0,0,0,0, ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=1, #A, m* ((1+x)^m - Ser(A))^m ), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 04 2019
STATUS
approved